The founding family of Nordstrom Inc. experienced their second setback in trying to take the company private on Monday, when a special committee of the board rejected their roughly $8 billion buyout offer as too low.

The buyout group, which includes private-equity firm Leonard Green & Partners, offered $50 a share, a price below the company’s latest closing price. As part of the deal, the group would also buy 21% of the family’s current stake, allowing the heirs to cash out about $550 million of their ownership in the...